Problem: Rewrite as a simplified fraction. ${0.\overline{47} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 47.4747...\\ 1x &= 0.4747...\end{align*}}$ Subtracting the two equations, we have: ${99x = 47}$ ${x = \dfrac{47}{99}} $